Chiral polytopes of order 2pᵐ

Let P be a chiral polytope with type ₁, k₂\ and G=Aut(P). Suppose |G|=2pᵐ, where k₁, k₂≥ 3 and p is an odd prime. Let P be a Sylow p-subgroup of G. We prove that G P Z₂, $d(P)=2$, P' ≠ 1(so m ≥ 3) and up to duality, ₁, k₂\=₁, 2pl₂\ for some integral l₁, l₂ ≥ 1. Moreover, we show that P is tight (k₁k₂=2pᵐ) if and only if P is metacyclic group. Furthermore, if $m=3$ or $4$, then P must be tight, and if m ≥ 5, where either m is odd, or m is even and m ≥ p+3, there exists a non-tight chiral polytope P.